Our paper is organized as follows. First we present some background in Section .
In the next section, Section , we collect some properties of the behavior of the functions from Theorem REF , near poles.
The first step of the proof of Theorem REF  is also done therein.
This part provides an upper estimate of \(\text{{\rm HD}}(\mathcal {I}(f))\)  which is a
straightforward adaption of the proof from [1]}. Indeed,
Bergweiler and Kotus used the most natural dynamical covers of the escaping set, in fact of \(\mathcal {I}_R(f)\) ,
from which the upper bound in Theorem REF  follows. The starting point of the present paper is the observation that their argument
gives in fact that
\(\text{{\rm HD}}(\mathcal {I}(f))\le \lim _{R\rightarrow \infty } \text{{\rm HD}}(\mathcal {I}_R(f))\le \delta .\) 
